The problem is that the potential scope and scale of a nuclear disaster, such as Chernobyl, is so huge that our minds don't really believe it. Our nuclear reactors in California have been operating for decades, and they have not melted down or blown up or spawned multi-headed livestock. Check out this article explaining why most California homeowners do not have earthquake insurance, despite the likelihood of a large quake: http://articles.moneycentral.msn.com/Insurance/InsureYourHome/why-you-may-be-in-disaster-denial.aspx
But the owners and operators of nuclear power plants, who would normally be on the hook if a reactor melted down, have insurance. And they have a federal law that limits their liability, and that provides for federal indemnification (read: taxpayer bailout) if the liability goes above that limit. It is called the Price-Anderson Act. http://www.eoearth.org/article/Price-Anderson_Act_of_1957,_United_States
Even that bastion of environmental protection, the Cato Institute, complains about how this is just one of the massive subsidies given to nuclear power: http://www.cato.org/pub_display.php?pub_id=3134
Oh, and the nuclear power industry gets government (taxpayer) loan guarantees, too: http://www.ucsusa.org/assets/documents/nuclear_power/nuclear-loan-guarantees.pdf
So we are subsidizing the nuclear power industry in a number of ways, including paying for part of their insurance, or more precisely, making it so that their insurers do not have to cover the full cost of a nuclear disaster - because we do.
If nuclear power was really so safe, would it need this protection against liability? If nuclear power was really so safe, would it need us to subsidize its insurance costs? If nuclear power was really so safe, why would it be asking us to pick up the tab if there is a disaster? There isn't really going to be a disaster, is there?